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Contrib Mineral Petrol (2015) 169:43DOI 10.1007/s00410-015-1138-3

ORIGINAL PAPER

Crystal structure, thermal expansivity, and elasticity of OH‑chondrodite: trends among dense hydrous magnesium silicates

Yu Ye1,2 · Steven D. Jacobsen3 · Zhu Mao4 · Thomas S. Duffy5 · Sarah M. Hirner2 · Joseph R. Smyth2

Received: 14 September 2014 / Accepted: 3 April 2015 / Published online: 19 April 2015 © Springer-Verlag Berlin Heidelberg 2015

velocities from which all 13 components (Cij) of the elas-tic tensor were determined. The Voigt–Reuss–Hill aver-age of the moduli yields for the adiabatic bulk modu-lus, KS0 = 117.9(12) GPa, and for shear modulus, G0 = 70.1(5) GPa. The Reuss bound on the isothermal bulk modulus (KT0) is 114.2(14) GPa. From the measured ther-modynamic properties, the Grüneisen parameter (γ) is cal-culated to be 1.66(4). Fitting previous static compression data using our independently measured bulk modulus (iso-thermal Reuss bound) as a fixed parameter, we refined the first pressure derivative of the bulk modulus, KT′ = 5.5(1). Systematic trends between H2O content and physical prop-erties are evaluated among dense hydrous magnesium sili-cate phases along the forsterite–brucite join.

Keywords Chondrodite · Dense hydrous magnesium silicates · Crystal structure · Thermal expansivity · Bulk modulus · Grüneisen parameter

Introduction

The dense hydrous magnesium silicate (DHMS) minerals along the brucite–forsterite join include phase A, norber-gite, chondrodite, humite, and clinohumite (Fig. 1) as well as other DHMS minerals in the peridotite system. Among these phases, phase A, chondrodite, and clinohumite have been observed as dehydration products of serpentine as well as in experiments on MgO–SiO2–H2O systems at pressures of 5–12 GPa (Berry and James 2001; Smyth et al. 2006; Wunder 1998). These phases could serve as H2O carriers in the Earth’s upper mantle and transition zone. Due to the critical role of water in controlling silicate phase relations and physical properties at upper mantle conditions, the crys-tal structures and physical properties of DHMS have been

Abstract We report the structure and thermoelastic prop-erties of OH-chondrodite. The sample was synthesized at 12 GPa and 1523 K, coexisting with hydroxyl-clinohu-mite and hydrous olivine. The Fe content Fe/(Fe + Mg) is 1.1 mol%, and the monoclinic unit-cell parameters are as follows: a = 4.7459(2) Å, b = 10.3480(7) Å, c = 7.9002(6) Å, α = 108.702(7)°, and V = 367.50(4) Å3. At ambient conditions, the crystal structure was refined in space group P21/b from 1915 unique reflection inten-sities measured by single-crystal X-ray diffraction. The volume thermal expansion coefficient was measured between 150 and 800 K, resulting in αV = 2.8(5) × 10−9 (K−2) × T + 40.9(7) × 10−6 (K−1) − 0.81(3) (K)/T2, with an average value of 38.0(9) × 10−6 (K−1). Brillouin spectroscopy was used to measure a set of acoustic

Communicated by Timothy L. Grove.

Electronic supplementary material The online version of this article (doi:10.1007/s00410-015-1138-3) contains supplementary material, which is available to authorized users.

* Yu Ye zylzylyeyu@gmail.com

1 State Key Laboratory of Geological Processes and Mineral Resources, China University of Geosciences, Wuhan 430074, Hubei, People’s Republic of China

2 Department of Geological Sciences, University of Colorado, Boulder, CO 80309, USA

3 Department of Earth and Planetary Sciences, Northwestern University, Evanston, IL 60208, USA

4 School of Earth and Space Sciences, University of Science and Technology of China, Hefei 230026, Anhui, People’s Republic of China

5 Department of Geosciences, Princeton University, Princeton, NJ 08544, USA

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extensively studied (e.g., Angel et al. 2001; McGetchin et al. 1970; Kanzaki 1991; Faust and Knittle 1994; Irifune et al. 1998; Kawamoto 2006; Komabayashi and Omori 2006; Ohtani et al. 2000; Sanchez-Valle et al. 2006; Shieh et al. 2000; Wunder et al. 1995; Ye et al. 2013). The com-position and various thermoelastic properties of phase A, chondrodite, and clinohumite are summarized in Table 1.

The humite minerals are found in metamorphosed lime-stones and dolomites and contain variable proportions of F and OH (Friedrich et al. 2002). Variable Ti and F/OH humites are also known in high-pressure ultramafic rocks (Evans and Trommsdorff 1983) as well as being recently described in a serpentinized wehrlite (Shen et al. 2014). Within kim-berlites, mineral associations including humites have indi-cated equilibrium pressures in excess of 6 GPa (Taskaev and Ilupin 1990). Synthetic hydroxyl-chondrodite (nomi-nally Mg5Si2O10H2) is stable up to at least 12 GPa at tem-peratures below 1473 K (Burnley and Navrotsky 1996), and Fe–Ti-bearing compositions bracket the stability fields of chondrodite and clinohumite in typical mantle compositions (Khodyrev et al. 1992). Fluorine may significantly increase its thermal stability (Engi and Lindsley 1980) and incompress-ibility (Friedrich et al. 2002), but the estimated concentration of fluorine in the upper mantle is limited (e.g., McDonough and Sun 1995). Hence, the geological occurrence of chon-drodite may be restricted to low-temperature regimes such as cold, subducting slabs, and the mantle wedge above them.

The crystal structure of chondrodite is illustrated in Fig. 2. The structure is monoclinic (space group P21/b, a-unique). The non-conventional setting of space group P21/c can be chosen to preserve the association with the a- and b-axes of olivine. In the crystal structure of OH-chon-drodite, there are three distinct (Mg,Fe)–O6 octahedra (M1, M2, and M3) and one Si–O4 tetrahedron (Si). The ratio of M1:M2:M3:Si equals 1:2:2:2, since the M1 sites are fixed at the special position of (1/2, 0, 1/2). Hydrogen atoms are bonded to O5 sites, which are also the positions for F ani-ons in natural F-bearing samples. Thermoelastic proper-ties of DHMS phases along the olivine–brucite join have been reported for a range of compositions (with variable H2O, Fe, Ti, F, etc.) including thermal expansivity (Pawley et al. 1995; Ye et al. 2013, etc.), isothermal compressibility

Fig. 1 Composition of hydrous magnesium silicates in the hydrous peridotite system (MgO–SiO2–Mg(OH)2). A: phase A; Nrb: norber-gite; Chn: chondrodite; Hu: humite; Chu: clinohumite; Fo: forsterite; En: enstatite; B: phase B; Ahy B: anhydrous phase B; Shy B: supper hydrous phase B; Tlc: talc; Ant: antigorite; E: phase E (non-stoichio-metric, Si/Mg 1.8–2.1, CH2O 6–18 wt%); D: phase D (Si/Mg 0.5–0.8, CH2O 13–19 wt%); 10 Å (CH2O 7–13 wt%); Hy Wad: hydrous wads-leyite (CH2O up to 3.3 wt%)

Table 1 Thermoelastic properties of phase A, chondrodite, and clinohumite (F-free) at ambient conditions

a Crichton and Ross (2002); b this study; c Ye et al. (2013); d Paw-ley et al. (1995); e Phan (2009); f refined on the data from Ross and Crichton (2001) and Kuribayashi et al. (2004); g Cynn et al. (1996); h Holland and Powell 1998; i the Debye temperatures and Grüneisen parameters are calculated in this study

Phase A Chondrodite Clinohumite

Formula Mg7Si2O14H6 Mg5Si2O10H2 Mg9Si4O18H2

CH2O (wt%) 11.8 5.4 2.9

ρ (g/cm3) 2.949(1)a 3.099(1)b 3.134(1)c

α0 (10−6 K−1) 46(2)d 38.0(9)b 36.8(6)c

KS0 (GPa) 100(1)e 118(1)b 119(2)e

G0 (GPa) 61(1)e 70.1(5)b 77(1)e

VP (km/s) 7.84(8)e 8.26(8)b 8.41(9)e

VS (km/s) 4.55(7)e 4.76(4)b 4.96(8)e

KT0 (GPa) 97.5(4)a 114(1)b 119.4(7)f

KT′ 6.0(1)a 5.5(1)f 4.8(2)f

CP (J kg−1 K−1) 924(1)g 874(1)g 884(1)h

Θac (K)i 726(13) 741(10) 760(15)

γi 1.69(8) 1.66(4) 1.50(4)

Fig. 2 Crystal structure of OH-chondrodite viewed down the a-axis, with hydrogen positions shown as black spheres. M1–M3 refer to (Mg, Fe) octahedra, and the silicate tetrahedron is labeled Si

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(Crichton and Ross 2002; Kuribayashi et al. 2004; Frie-drich et al. 2002; Holl et al. 2006; Ross and Crichton 2001), and adiabatic elastic moduli (Fritzel and Bass 1997; Jacob-sen et al. 2008; Jiang et al. 2006; Sanchez-Valle et al. 2006; Sinogeikin and Bass 1999; Phan 2009; Xia et al. 1998). In this study, we examined the crystal structure of a Fe-bear-ing, OH-chondrodite sample by single-crystal X-ray dif-fraction and determined the lattice coefficients of thermal expansivity between 150 and 800 K. Brillouin spectroscopy was used to measure a set of velocities from three differ-ent platelets that constrain the thirteen independent Cij of the elasticity tensor. By comparing the current results with physical properties of other DHMS phases along the forst-erite–brucite join, we establish trends as a function water content that can be used to model the physical properties of mineral assemblages found in hydrated subducting slabs.

Experiments

Sample synthesis and composition

The Fe-bearing OH-chondrodite crystals were synthesized in the 5000-ton multi-anvil press at Bayerisches Geoinsti-tut, Bayreuth, Germany, run SZ0407B (Smyth et al. 2006). An outer 3.5-mm welded Pt capsule contained two separate 1.6-mm-diameter welded Pt inner capsules in a matrix of brucite to maintain H activity. The inner capsule contained a 1-mm single-crystal cube of San Carlos olivine, plus a mixture of brucite and forsterite. The inner capsules also contained small amount of a Re–ReO2 oxygen buffer to maintain the iron in the ferrous state. The run was made in an 18-mm sintered MgO octahedron using 54-mm WC cubes with 8-mm corner truncations. The nominal pressure was 12 GPa at 1250 °C with a heating duration of 180 min. The original objective of the experiment was to hydrate the San Carlos olivine, but rupture of the inner capsule exposed the San Carlos olivine to excess brucite, with which it reacted to form phase A, chondrodite, and clinohumite. The chondrodite crystals were up to 0.25 mm in diameter and light tan in color.

A single crystal about 120 µm in size was selected and polished for chemical analysis by electron microprobe fol-lowing the experimental procedure described previously for the coexisting clinohumite (Ye et al. 2013). Six points at different locations of the crystal were chosen for analysis, and the average values together with their standard devia-tions are listed in Table 2. The water content was calcu-lated to be 5.26 wt%, assuming the molar ratio of H:O to be 2:10. The formula is expressed as Mg5.047(4)Fe0.058(2)

Ti0.0001Mn0.0004(1)Ni0.0018(5)Al0.0017(2)Ca0.0008(2)Si1.945(3)

O10H2, and Fe mol% = Fe/(Fe + Mg) = 1.1 % which is lower than those of the coexisting OH-clinohumite (4 % Fe) (Ye et al. 2013) and hydrous olivine (3 % Fe) (Jacobsen et al. 2008, hy-Fo97) in the run products.

Single‑crystal X‑ray diffraction (XRD)

A single crystal of OH-chondrodite measuring 100 × 85 × 80 µm3 was chosen for X-ray diffraction at ambient conditions. Unit-cell parameters were refined on a Bruker P4 four-circle diffractometer with a dual-scin-tillation point detector system using an 18-kW rotating Mo-anode X-ray source operating at 50 kV and 250 mA. The average wavelength for MoKα1–Kα2 mixed char-acteristic wavelengths was determined to be 0.71073 Å by measuring a spherical single crystal of anhydrous for-sterite (Fo100). Least squares fitting was performed on 48 centered reflections within the following classes: (120), (11̄2), (12̄2), (13̄1), (130), (112), (13̄2), (131), (210), (11̄3), (12̄3), (122), (14̄1), (211), (21̄2), (22̄3), (222), and (24̄1), and the refined unit-cell parameters at ambient condi-tion are as follows: a = 4.7459(2) Å, b = 10.3480(7) Å, c = 7.9002(6) Å, α = 108.702(7)°, and V = 367.50(4) Å3. Intensity data were collected using a Bruker APEX II CCD detector mounted on a P4 diffractometer up to 2θ = 75°. Refinement of atomic positions and anisotropic displace-ment parameters was carried out using SHELXL-97 (Sheldrick 2008) in the software package WinGX (Far-rugia 2012), with the default neutral atom scattering fac-tors in SHELXL-97. In total, 10,333 reflections were collected with 1915 unique and 1833 with I > 4σ used

Table 2 Composition of OH-chondrodite SZ0407B from electron microprobe analyses

a Atoms per formula unitb The wt% and apfu values for H2O/H are calculated from stoichiometry, assuming that the molar ratio of H:O is 2:10

wt% apfua wt% apfu

SiO2 34.09(6) Si 1.945(3) NiO 0.04(1) Ni 0.0018(5)

TiO2 0.003(3) Ti 0.0001(1) MnO 0.008(3) Mn 0.0004(1)

Al2O3 0.026(3) Al 0.0017(2) CaO 0.014(3) Ca 0.0008(2)

MgO 59.34(5) Mg 5.047(4) H2Ob 5.26 H 2

FeO 1.22(3) Fe 0.058(2) Total 100.00(6)

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in the refinement. Model fit parameters are as follows: R1 = 0.0213 for I > 4σ, and 0.0223 for all the unique reflec-tions; goodness of fit (GooF) = 1.674; and for merging of equivalent reflections, Rint = 0.039.

The refined atomic position coordinates and occupancy factors are listed in Table 3, and the anisotropic displace-ment parameters are given in “Appendix 1”. In this study, Fe and Mg occupancies in the M1, M2, and M3 sites were refined, as well as Si occupancy, and the O occupancies were fixed at full occupancy. In addition, the H position was refined with an isotropic displacement parameter (Ueq) fixed at 0.054.

The same single crystal of OH-chondrodite was then used for thermal expansion measurements below and above room temperature. At each temperature step, the refine-ment of unit-cell parameters was performed using the same experimental procedure as the room-temperature measure-ment described above. Low-temperature measurements were taken at 253, 203, and 153 K. Temperatures were measured and controlled by a Bruker LT-2A controller and N2 gas stream nozzle. For high-temperature measurements, the crystal was transferred to a silica glass capillary, sealed at both ends. Ten high-temperature steps were carried out from 350 to 787 K using a Bruker high-temperature device with a two-prong ceramic-coated Pt wire radiant heater and an Omega temperature control unit (Ye et al. 2009). The refined unit-cell parameters at low and high temperatures are given in “Appendix 2”.

Brillouin spectroscopy

Three crystals of OH-chondrodite (from run SZ0407B) were prepared for Brillouin scattering experiments by double-sided polishing to produce parallel plates

measuring about 100 µm across and 30–50 µm thick. Pol-ishing was carried out with 3-µm diamond lapping film and finishing with 1 µm and then finally 0.5-µm diamond lapping film to produce an optical-quality surface free of visible scratches.

Prior to Brillouin spectroscopy, Raman spectroscopy was carried out on each of the three platelets. Unpolarized Raman spectra were obtained using a 200-mW Ar-ion laser (λ = 514.532 nm) with a confocal optical setup in back-scattering geometry using holographic optics and a single-grating 0.5-m spectrometer with a 1100 × 330 pixel CCD camera. Raman spectra of the three OH-chondrodite crys-tals used in the Brillouin scattering experiments are shown in Fig. 3a–d.

Brillouin spectra were obtained in a forward symmetric scattering geometry (Speziale and Duffy 2002; Speziale et al. 2014) with a six-pass Sandercock tandem Fabry–Perot interferometer, using a solid-state laser (λ = 532.15 nm, 150 mW at the sample). Acoustic velocities (Vi) were deter-mined as:

where Δωi is the Brillouin scattering frequency shift of the acoustic modes, and θ is the scattering angle in the sample (70° in this study). For each platelet, measurements were taken in ~5–10° steps with the total number of spectra in a given platelet depending on its size and the quality of the signal. Measured acoustic velocities are plotted as func-tions of azimuthal angle in Fig. 4a–c. Further details about the Brillouin scattering measurements can be found in Spe-ziale and Duffy (2002) and Speziale et al. (2014).

Three platelets were used for Brillouin scattering, with the orientation of each platelet defined with three Eulerian angles (θ, φ, and χ). Chondrodite has a monoclinic struc-ture with thirteen elastic constants: C11, C22, C33, C44, C55, C66, C12, C13, C23, C15, C25, C35, and C46. The elastic con-stants and acoustic velocities are related by Christoffel’s equation (Every 1980):

where Cijkl is the elastic tensor in full suffix notation, ni is the direction cosine of the phonon propagation direction, density ρ = 3.099 g/cm3 (according to the composition determined by EPMA and unit-cell volume from single-crystal XRD at ambient condition), and VP,S is the acoustic velocity of either longitudinal (VP) or transverse (VS) in the Brillouin spectra. The inversion was carried out by nonlinear least squares using Levenberg–Marquardt method (Press et al. 1988) with closed-form expressions for the Christoffel equation given

(1)Vi =�ωi�

2 sin(

θ2

) ,

(2)det |Cijklnjnl − ρV2P,Sδik| = 0

Table 3 Atomic positions and occupancy factors from X-ray inten-sity scan (space group: P21/b)

Atom x/a y/b z/c Occupancy

M1 0.5 0 0.5 Mg: 0.991(2)/Fe: 0.009(2)

M2 0.00761(6) 0.17542(3) 0.30892(4) Mg: 0.992(2)/Fe: 0.008(2)

M3 0.48780(6) 0.88398(3) 0.07707(4) Mg: 0.989(2)/Fe: 0.011(2)

Si 0.07769(4) 0.14073(2) 0.70194(2) 0.983(2)

O1 0.77577(11) 0.00294(5) 0.29661(7) 1

O2 0.72759(11) 0.24411(5) 0.12853(7) 1

O3 0.22561(11) 0.16797(5) 0.52799(7) 1

O4 0.26251(11) 0.85683(5) 0.29738(7) 1

O5 0.26372(14) 0.06017(6) 0.10540(9) 1

H 0.109(3) 0.046(2) 0.004(3) 1

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by Every (1980). The inversion process was iterative. The elastic constants of Sinogeikin and Bass (1999) were used as the initial model and held fixed, while the inversion was per-formed for the three Eulerian angles defining the orientation of each plane. The relationship between the Eulerian angles and the direction cosines of the phonon propagation direc-tion is given by Shimizu (1995). In subsequent inversions, the elastic constants and orientations were alternately var-ied or allowed to vary together, and the process was contin-ued until convergence was achieved. Different sets of initial elastic constants were tested to confirm that the final results were insensitive to the starting model. The results are shown in Fig. 4a–c. The good quality of the fit is demonstrated by the small root-mean-square difference between the measured and calculated velocities which was VRMS = 39 m/s.

Results and discussion

Raman spectroscopy

Unpolarized Raman spectra of the three OH-chondrodite platelets used for Brillouin scattering are shown in Fig. 3. The spectra are generally similar to previous Raman stud-ies of OH-chondrodite (Lin et al. 1999; Mernagh et al. 1999). In Fig. 3, the polished plane orientation is expressed as (hkl), fitted from the velocity dispersion curves in the Brillouin scattering study. Band assignments for some of the modes shown in Fig. 3 are given elsewhere (Cynn et al. 1996; Lin et al. 1999). In the O–H stretching region, we find a similar strong doublet as was found for F-free syn-thetic OH-chondrodite (Lin et al. 1999). The main band

Fig. 3 Unpolarized Raman spectra of the three OH-chondrodite crys-tals used in the Brillouin scattering experiments: a low-wavenumber region, b mid-wavenumber region, c high-wavenumber region show-ing O–H stretching bands, d high-wavenumber region of one sample shown with the two main O–H stretching bands deconvolved into six

different components (dashed lines), sum of the fitted peaks (solid line) and spectral data points (gray dots). The unpolarized Raman spectra were collected in backscatter geometry with the incident laser perpendicular to each plane expressed as (hkl), fitted by Brillouin spectroscopy

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positions shown in Fig. 3 are 3525 and 3565 cm−1, which are shifted by about 10 cm−1 from the positions shown by Lin et al. (1999) at 3515 and 3554 cm−1. Also in Fig. 3 is shown a deconvolution of the O–H stretching doublet (dashed lines), which is composed of at least four separate bands at 3525, 3560, 3565, and 3580 cm−1. The 3525 cm−1 band may be further deconvolved into three separate bands to best fit the observed spectrum.

Crystal structure

According to the refined Mg and Fe occupancies in Table 3, the total Fe mol% is 0.9(4) %, consistent within uncertainty with that of (1.1 mol%) obtained from electron microprobe analyses. The Fe content from single-crystal XRD (site occupancy) is an approximation as minor elements (Ti, Ni, Mn, Ca, Al, < 0.04 mol%) were ignored in the structural refinement. The Si occupancy refinement from X-ray dif-fraction is consistent with that from microprobe analyses. The bond lengths and polyhedral volumes were calculated using the software package XTALDRAW (Downs et al. 1993) and listed in Table 4. The octahedral volumes show the order of M2 > M3 > M1, and the Si tetrahedral vol-ume is 2.21(1) Å3. The hydrogen bond geometry shows an O5–H distance of 1.06(2) Å and O5···H of 2.12(2) Å with a hydrogen bond angle of 122.7˚. The O5–H bond length (O/F–H bonds in some chondrodites) has been refined by neutron diffraction (Friedrich et al. 2001 and Kunz et al. 2006), as well as X-ray diffraction (Ottolini et al. 2000 and Friedrich et al. 2002). Neutron diffraction better reflects the OF5–H interatomic distances than single-crystal X-ray dif-fraction. Friedrich et al. (2001) reported a bond length of 1.029(6) Å for OF–H and 1.872(6) Å for OF5···H, while Kunz et al. (2006) reported 0.976(7) and 1.940(7) Å, respectively. Our O5–H bond length is close to those from neutron experiments, whereas the O5···H interatomic dis-tance is 10 % longer than previous results from neutron diffraction.

To compare the compositional effects of Fe, F, and Ti on the crystal structure of chondrodite, the unit-cell volumes and M–OF bond lengths are summarized in Table 5 for this study (SZ0407B) and literature values with the follow-ing abbreviations used throughout: Berry and James 2006, B02; Friedrich et al. 2001, 2002, F01 and F02; Fujino and Takéuchi 1978, F78; Gibbs et al. 1970, G70; Kuribayashi et al. 2004, K04; Kunz et al. 2006, K06; Ottolini et al. 2000, O00; Yamamoto 1977, Y77. The α angle in sample F78 is significantly larger than other samples in Table 5, which is likely due to the effect of Ti4+ incorporation

Fig. 4 a–c Longitudinal acoustic (LA) and transverse acoustic (TA) velocities in three different planes (hkl) of OH-chondrodite as a func-tion of crystallographic direction. The azimuthal angle is relative to an arbitrary starting angle. The solid curves were calculated using the best-fit elastic constants for comparison with the measured velocities (filled circles)

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(Robinson et al. 1973). In addition, α angle values in SZ0407B and Y77 are 0.35–0.4° smaller than those in Y77, B02, G70, O00, F01, F02, K06, and K04, implying that F− might increase the angle. The unit-cell volumes decrease in the order of SZ0407B and Y77 > F78 > F01, F02, K06 and K04 > B02 and G70 > Y77 consistent with the order of decreasing ionic radii (Fe2+ > Mg2+ > Ti4+) and anions (O2− > F−). Samples SZ0407B, Y77, and F78 are F-free, while F78 has a high Ti4+ concentration of 8.4 mol%. Samples F01, F02, K06, and K04 have higher Fe2+ concentrations than B02, G70, and Y77, and B02 has the highest F− content among these samples. The

most significant effect of F− on the crystal structure is that F− decreases distances between OFOH and surrounding cations, i.e., one M2-OF bond and two M3-OFOH bonds. Samples SZ0407B and Y77 typically have larger M-OF bond lengths than other samples, as shown in Table 5, while sample F78 has significantly smaller M3-OF bond lengths, due to the occupancies of smaller cations Ti4+ in M3 site. Fe2+ should increase the M-O bond lengths, but such effect is not well observed here, perhaps because of the low concentration of Fe2+ in the reported crystal struc-tures and, possible technical discrepancies among different measurements.

Table 4 Bond lengths (Å) and polyhedral volumes (Å3)

M1 M2 M3

O1 × 2 2.0805(5) O1 2.0713(6) O1 2.2413(6)

O3 × 2 2.1253(5) O2 2.2257(6) O2 2.0140(6)

O4 × 2 2.1227(5) O3 2.0387(6) O2 2.1123(6)

Avg. bond 2.1095(5) O3 2.1772(6) O4 2.1363(6)

Poly. V (Å3) 12.04(1) O4 2.1983(6) O5 2.0604(7)

O5 2.0630(7) O5 2.0819(7)

Avg. bond 2.1290(7) Avg. bond 2.1077(7)

Poly. V (Å3): 12.47(1) Poly. V (Å3): 12.13(1)

Si H

O1 1.6461(6) O5 1.06(2)

O2 1.6427(6)

O3 1.6463(6)

O4 1.6151(6)

Avg. bond 1.6376(6)

Poly. V (Å3) 2.22(1)

Table 5 Selected bond length (Å) for chondrodite samples from this and previous studies

SZ0407B: this study; Y77: Yamamoto (1977); B02: Berry and James (2006); G70: Gibbs et al. (1970); O00: Ottolini et al. (2000); F01: Friedrich et al. (2001); F02: Friedrich et al. (2002); K06: Kunz et al. (2006); K04: Kuribayashi et al. (2004): F78: Fujino and Takéuchi (1978)a Atomic percentage in the total metallic cationsb Molar ratio of F/(F + OH)c Synthesized from experiments, while others are from natural sourcesd Data measured at ambient condition

Fe%a F%b Ti%a α (°) V (Å3) M2-OF5 M3-OF51 M3-OF52

SZ0407Bc 1.1 0 0 108.702(7) 367.50(4) 2.0628(5) 2.0599(6) 2.0825(6)

Y77c 0 0 0 108.71(5) 368.7(2) 2.133(3) 2.056(3) 2.091(3)

B02 1.4 77 0 109.11(1) 357.26(3) 2.054(8) 2.011(8) 2.015(9)

G70 1 65 0 109.059(2) 359.30(1) 2.045(2) 2.016(1) 2.042(2)

O00 0.8 62 0 109.08(2) 359.47(1) 2.051(3) 2.019(3) 2.045(3)

F01d 5.6 51 0.46 109.097(2) 363.14(4) 2.060(1) 2.033(1) 2.059(1)

F02d 5.6 58 0.46 109.065(1) 361.98(2) 2.057(8) 2.01(1) 2.055(9)

K06d 5.6 58 0.46 109.065(3) 361.98(2) 2.068(5) 2.048(5) 2.067(5)

K04d 4.4 37 0.4 109.07(1) 361.0(2) 2.057(1) 2.024(2) 2.053(1)

F78 12.5 0 8.4 109.33(1) 363.83(5) 2.051(2) 1.976(2) 1.987(2)

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Thermal expansivity

To relate the anisotropy of thermal expansion to the crystal structure, the principal axes of thermal expansion are deter-mined for this monoclinic structure, reducing the number of thermal expansion coefficients from 4 to 3: X1, X2, and X3 (Cliffe and Goodwin 2012). The variations with tempera-ture of the principle expansion axes are listed in “Appendix 2”, and the average axial thermal expansion coefficients are listed in Table 6, as well as the orientations of the principle expansion axes. There is no difference between a and X1 for a-unique monoclinic structure, while the thermal expan-sion coefficient of X2 is 8 % smaller than that of b-axis for the compensation of α angle (>90˚) expansion.

The thermal expansion coefficient is defined as:

Fei (1995) proposed a useful formula for modeling α(T) above room temperature:

On the basis of the Mie-Grüneisen-Debye equation of state, Suzuki (1975), Suzuki et al. (1979), and Kumar (1995, 1996, 2003) developed expression for the unit-cell volume as function of temperature:

where k = (KT′ − 1)/2 = 2.25 (KT′ = 5.5 as in Table 1), Q0 = V0(0)KT0(0)/γMGD. V0(0) and KT0(0) are the volume and isothermal bulk modulus taken at 0 K and 0 GPa. γMGD is a parameter related to the thermodynamic Grüneisen parameter γth (Barron et al. 1980). The thermal energy Eth(T) is expressed by the Debye model:

(3)α(T) =1

V

(

∂V

∂T

)

P

=

(

∂ ln (V)

∂T

)

P

.

(4)α(T) = a0 + a1T +a2

T2(Fei equation).

(5)

V0(T) = V0(0) ·

[

1+1− (1− 4k · Eth(T)/Q0)

1/2

2k

]

(Suzuki equation)

(6)

V0(T) = V0(0) ·

[

1−ln[1− (2k + 2) · Eth(T)/Q0]

1/2

2k + 2

]

(Kumar equation)

where R is the gas constant, n is the number of atoms in the formula unit (19 for OH-chondrodite), and ΘD is the Debye temperature, and the acoustic Debye temperature Θac is given by:

where h, k, N are Boltzmann, Plank, and Avogadro’s con-stants, respectively, M is the molar mass, and Vm is the mean velocity. The calculated Θac values for phase A, chondrodite, and clinohumite are listed in Table 1. The Θac values for these DHMS phases are close to that (729 K) for San Carlos olivine (Liu and Li 2006), and we adopt ΘD = 741 K to calculate the thermal expansion coefficient of chondrodite in this study.

The unit-cell volume and principal expansion axes are plotted in Fig. 5a as a function of temperature and fitted by the curves calculated from Fei equation (Eq. 4). The principal expansion axes show an order of αX3 > αX1 > αX2. We derive the thermal expansion coefficients for unit-cell volume as follows: a0 = 40.9(7) × 10−6 K−1, a1 = 2.8(5) × 10−9 K−2, a2 = − 0.81(3) K according to Fei equation. Next, the V(T) dataset from this study is also fitted by Suzuki and Kumar equations (Eqs. 5, 6). We used KT0

(298 K) = 114 GPa (Table 1), and assuming that OH-chon-drodite has the same dKT0/dT as forsterite (−0.023 GPa/K, Kroll et al. 2012), we obtain KT0 (0 K) = 121 GPa. Then, the fittings result in γMGD = 1.19(6) from Kumar equation; 1.21(8) from Suzuki equation; and V0 (0 K) = 365.8(3) Å3 from both Kumar and Suzuki equations.

The calculated αV (T), calculated from Fei, Kumar, and Suzuki equations, are plotted in Fig. 5b, respectively, and extrapolated to 1500 K where the breakdown of OH-chon-drodite was observed (Burnley and Navrotsky 1996). The

(7)Eth(T) =9nRT

(�D/T)3

�D/T∫

0

x3

ex − 1· dx

(8)�ac =

h

k·

(

3N

4π

)1/3

·

(n · ρ

M

)1/3

· Vm

(9)3

V3m

=

2

V3s

+

1

V3p

Table 6 Average thermal expansion coefficients for the unit-cell parameters and the principal expansion axes (153–787 K), and the orientations of the principal axes

Unit-cell parameters Principal expansion axes Orientations

α0 (10−6 K−1) α0 (10−6 K−1) a b c

a 12.7(3) X1 12.7(3) 1 0 0

b 11.5(2) X2 10.6(2) 0 0.8353 0.5499

c 14.6(4) X3 14.7(4) 0 −0.1636 0.9865

α 1.43(10)

V 38.0(9)

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Fei equation generates negative αV below 140 K, whereas the Kumar and Suzuki equations give larger and more reason-able αV at such low temperatures. From 200 to 800 K, these three equations agree with each other well, with discrepancy of 5 % at 800 K. When extrapolated to 1500 K, the Kumar and Suzuki equations indicate αV values 20 and 13 % larger, respectively, compared with that from Fei equation. In addi-tion, for α (T) relation as proposed by Fei (1995), the sum of the thermal expansion coefficients for principal axes at the measured temperatures is also plotted in Fig. 5b, which con-sistent with the solid curve of αV(T) from Fei equation.

The axial thermal expansivity for the present Fe-bearing OH-chondrodite sample shows anisotropy in the order of αc > αa > αb, whereas axial compressibilities determined from single-crystal XRD data at high pressure yield an

anisotropy of βb ≈ βc > βa for Fe-free OH-chondrodite (Ross and Crichton 2001; Kuribayashi et al. 2004) and βb > βc > βa for Fe,F-bearing samples (Friedrich et al. 2002; Kuribayashi et al. 2004). The axial compressibilities are also consistent with the elasticity studies: C11 > C33 ≈ C22 for Fe-free OH-chondrodite (this study, in the following discussion) and C11 > C33 > C22 for Fe,F-bearing sample (Sinogeikin and Bass 1999). Here, the elastic moduli are transferred to a-unique monoclinic symmetry to be consist-ent with the studies of crystal structure and isothermal com-pression. In this study, we find that the α angle increases with temperature, which is consistent with the observations that α decreases with pressure (Ross and Crichton 2001; Friedrich et al. 2002; Kuribayashi et al. 2004). The angle α increases as temperature increasing with a thermal expan-sion coefficient of 1.43(10) × 10−6 K, consistent with that for clinohumite (1.41–1.53 × 10−6 K from Ye et al. 2013).

The average volume thermal expansion coefficients (α0) for the DHMS phases along the forsterite–brucite join are plotted in Fig. 6, as a function of water content (CH2O, expressed as wt% H2O). The α0 (V) values are recalcu-lated over temperature range of 300–787 K for forsterite (Ye et al. 2009), clinohumite (Ye et al. 2013), and chon-drodite (this study) to allow for consistent comparison with previous experiments on phase A (Pawley et al. 1995) and brucite (Fukui et al. 2003). There have been many previ-ous studies on thermal expansivity of anhydrous forster-ite (e.g., Ye et al. 2009; Kroll et al. 2012; and Trots et al. 2012). To simplify Fig. 6, we plotted the anhydrous forst-erite thermal expansivity data from Ye et al. (2009) as rep-resentative. The α0 for hydrous forsterite (0.89 wt% H2O)

Fig. 5 a Unit-cell volume and principal expansion axes of OH-chon-drodite as a function of temperature between 153 and 787 K. The solid curves are fitting to the data normalized to 153 K, using the Fei equation. Experimental uncertainties are smaller than the size of the symbols. b Volumetric thermal expansion coefficient as a function of temperature from Fei, Suzuki, and Kumar equations, which are extrapolated to 1500 K

Fig. 6 Average thermal expansion coefficients as a function of water content for DHMS phases along the forsterite–brucite join. The tem-perature ranges are 300–787 K for forsterite (Fo, Ye et al. 2009), clinohumite (Chu, Ye et al. 2013), and chondrodite (Chn, this study); 293–873 K for phase A (PhA, Pawley et al. 1995); 299–593 K for brucite (Bru, Fukui et al. 2003)

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is ~0.8 × 10−6 K−1 larger than that for anhydrous forster-ite. The Fe-bearing clinohumite (Fe% = 4.1 mol%) shows α0 ~ 2.2 × 10−6 K−1 smaller than Fe-free clinohumite (Ye et al. 2013). The α0 for the current OH-chondrodite sample is 40.8(4) × 10−6 K−1 in the range of 300–787 K, whereas it is 38.0(9) × 10−6 K−1 in 153–787 K.

A linear regression for the data of Fig. 6 yields a system-atic relation between α0 and CH2O as:

Each additional 1 wt% H2O content increases α0 by 1.02(3) × 10−6 K−1 (or about 3 %). Thus, the thermal expansion coefficient increases with H2O content for

(10)α0 (10

−6K−1) = 35.3(4)+ 1.02(3)

× CH2O(wt%) (R2 = 0.9947)

DHMS along the forsterite–brucite join, which is consist-ent with the observation that the bulk moduli decrease as H2O content increases (Ross and Crichton 2001; Holl et al. 2006; Jacobsen et al. 2008; Sanchez-Valle et al. 2006).

After heating to 787 K, the single crystal of OH-chon-drodite was quenched to room temperature, and the unit-cell parameters after heating are as follows: a = 4.7455(4) Å, b = 10.3475(8) Å, c = 7.9008(9) Å, α = 108.706(8)°, and V = 367.47(7) Å3, consistent with the initial ones before heating. We conclude that this Fe-bearing OH-chondrodite sample persisted to 787 K at ambient pressure without any observable phase transition or dehydration.

Elastic moduli

The calculated elastic moduli for OH-chondrodite (Fe/(Mg + Fe) = 1.1 %) are listed in Table 7, in comparison with F-bearing chondrodite (Fe/(Mg + Fe) = 5.5 %, F/(F + OH) = 32 %, Sinogeikin and Bass 1999). Nearly all of the longitudinal and shear elastic constants are lower than those reported by Sinogeikin and Bass (1999) for a F-rich natural chondrodite. The Brillouin inversion was performed in standard monoclinic symmetry (b-unique, P21/c) where zigzag chains of edge-sharing octahedra are aligned along the a-axis and lie in the plane perpendicular to the b-axis. The elastic indices are also reported in the a-unique coordi-nate system (P21/b) (values given in parentheses in Table 7) to facilitate comparison with the studies of the crystal struc-ture. The longitudinal constants C22 and C33 (in standard monoclinic symmetry hereafter) are close to the values from Sinogeikin and Bass (1999), whereas C11 is 12 % smaller. The shear constants C55 and C66 are about 5 % smaller, while C44 is 8 % smaller. The off-diagonal moduli are not

Table 7 Single-crystal elastic constants (in GPa) for chondrodite at ambient conditions

SB89: Sinogeikin and Bass (1999)

The elastic constants are given in the standard monoclinic coordinate system (b-unique, P21/c), while the values inside parentheses are those for an a-unique coordinate system (P21/b)

This study SB89 This study

SB89

Longitudinal Off-diagonal

C11(C33) 188.0(7) 213.4(15) C12(C13) 61(10) 70(3)

C22(C11) 278.0(10) 275.3(15) C13(C23) 71.4(7) 59(2)

C33(C22) 195.7(7) 198.4(12) C23(C12) 74(3) 67(3)

Shear C15(C14) 3.9(4) 7.2(10)

C44(C66) 64.0(10) 69.7(6) C25(C24) 2.6(10) −1.7(12)

C55(C44) 68.3(4) 72.1(9) C35(C34) 2.5(4) −2.6(8)

C66(C55) 71.3(5) 75.2(7) C46(C56) 4.1(6) −0.7(4)

Table 8 Aggregate elastic moduli and pressure derivative of bulk modulus for chondrodite

a Adiabatic bulk modulusb KT0 calculated from Reuss bound KS0c Refined on the datasets of Ross and Crichton (2001) and Kuribayashi et al. (2004), with KT0 = 114.2 GPad Fixed without refinement

F/(F + OH) Fe (mol%) KT0 (GPa) KT′ G0 (GPa)

This study 0 1.1 119.4a 71.1 (Voigt bound)

116.4a 69.2 (Reuss bound)

117.9(12)a 70.1(5) (VRH average)

114.2b 5.5(1)c

Sinogeikin and Bass (1999) 0.32 5.5 118.4(16)a 75.6(7)

Ross and Crichton (2001) 0 0 115.7(8) 4.9(2)

Kuribayashi et al. (2004) 0 0 117(2) 4d

Kuribayashi et al. (2004) 0.37 4.4 124.1(4) 4d

Friedrich et al. (2002) 0.55 0 117(4) 5.6(1)

Faust and Knittle (1994) 0.69 0 136.2 3.7d

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constrained as well in either study. The Voigt–Reuss bounds and Voigt–Reuss–Hill (VRH) average of the aggregate moduli (Watt 1980) for OH-chondrodite are summarized in Table 8. The bulk modulus found here is identical within uncertainty to that obtained by Sinogeikin and Bass (1999)

and consistent with those reported in static compression studies (Ross and Crichton, 2001, Kuribayashi et al., 2004; Friedrich et al., 2002). The shear modulus, G0, is, however, about 6 % lower than that reported by Sinogeikin and Bass (1999) for a natural F-bearing sample, indicating that the shear modulus may increase with F content.

Using the density (ρ), isothermal bulk modulus (KS0), thermal expansion coefficient (α0), and heat capacity (CP) listed in Table 1, we calculated the Grüneisen parameter (γ) for phase A, OH-chondrodite, and OH-clinohumite, from the formula:

The calculated γ values are also listed in Table 1. The Grüneisen parameter for phase A [1.69(8)], chondrodite [1.66(4)], and clinohumite [1.50(4)] is larger than that for forsterite (1.28, Gillet 1991), since phase A, chondrodite, and clinohumite have smaller CP and ρ values than forster-ite, whereas variation in the product α·KS is relatively less significant among these phases.

Because the Reuss bound on the bulk modulus, KS0, cor-responds to constant stress across grain boundaries, which is relevant for static compression studies, we calculate the isothermal Reuss bound bulk modulus for OH-chondrodite: KT0 = 114.2(14) GPa, with KS0 = 116.4(12) GPa and the formula:

(11)γ =α · KS

ρ · CP

Table 9 Studies of elasticity for DHMS along the olivine–brucite join, listed in the order of H2O content increasing

a F/(F + OH) = 0.29 for clinohumiteb F/(F + OH) = 0.32 for chondrodite

H2O (wt%) Density (g/cm3) Fe (mol%) KS0 (GPa) G0 (GPa) VP (km/s) VS (km/s)

Olivine

Suzuki et al. (1983) 0 3.225(1) 0 128.9(2) 81.4(2) 8.58(1) 5.02(1)

Zha et al. (1996) 0 3.221(1) 0 128.8(5) 81.6(2) 8.59(3) 5.03(1)

Webb (1989) 0 3.350(1) 10 129.5(2) 77.5(2) 8.34(1) 4.81(1)

Zha et al. (1998) 0 3.343(1) 10 131.1(19) 79.4(8) 8.42(9) 4.87(4)

Jacobsen et al. (2008, 2009) 0.8 3.240(1) 3 125.2(2) 77.7(1) 8.40(1) 4.90(1)

Jacobsen et al. (2008, 2009) 0.9 3.180(1) 0 125.4(4) 79.6(3) 8.53(2) 5.00(2)

Clinohumite

Fritzel and Bass (1997) 2.0a 3.261(1) 6.5 125(2) 73(5) 8.26(9) 4.7(3)

Phan (2009) 2.9 3.134(1) 0 119(2) 77(1) 8.41(9) 4.96(8)

Chondrodite

Sinogeikin and Bass (1999) 3.3b 3.227(1) 9.4 118(2) 75.6(7) 8.24(8) 4.84(5)

This study 5.3 3.099(1) 1.1 118(1) 70.6(5) 8.26(8) 4.76(4)

Phase A

Sanchez-Valle et al. (2006) 11.7 2.976(1) 1.9 106(1) 61(1) 7.93(8) 4.53(8)

Phan (2009) 11.8 2.949(1) 0 100(2) 61(1) 7.84(9) 4.55(8)

Brucite

Jiang et al. (2006) 30.9 2.380(1) 0 43.8(8) 35.2(3) 6.2(1) 3.85(3)

Xia et al. (1998) 30.9 2.380(1) 0 46(1) 34.9(5) 6.2(1) 3.83(5)

1.00

0.99

0.98

0.97

0.96

0.95

0.94

V/V

0

876543210

P (GPa)

Ross and Crichton (2001) Kuribayashi et al. (2004) 3rd-order BM EoS fittingKT0 = 114.2 GPa, K' = 5.5(1)

Fig. 7 Unit-cell volumes of Fe-free OH-chondrodite samples (Ross and Crichton 2001; Kuribayashi et al. 2008) as a function of pres-sure, with third-order Birch–Murnaghan equation of state (BM EoS) fitting. The volumes are normalized to their respective V0 values at ambient conditions

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Next, on the basis of reported compression datasets for Fe-free OH-chondrodite (Ross and Crichton 2001; Kurib-ayashi et al. 2004), we refined the pressure derivative of the bulk modulus, KT′ and obtain KT′ = 5.5(1) (Fig. 7) by fix-ing KT0 at 114.2 GPa, the Reuss bound from Brillouin scat-tering. Figure 7 also shows that both these high-pressure datasets are consistent with each other. It should be noted that our OH-chondrodite has 1.1 mol% Fe, but such a low Fe concentration should probably be expected to have lit-tle effect on the bulk modulus. The refined results are also included in Table 8. Our results are consistent with the chondrodite sample from Friedrich et al. (2002) (Fe-free, F/(F + OH) = 0.32) which further indicates that F content does not strongly affect the bulk modulus. It should also be noted that the bulk modulus obtained by Faust and Knittle (1994) could be affected by non-hydrostatic stress due to their use of a methanol–ethanol–water pressure-transmit-ting medium at pressures up to 42 GPa.

Over the past three decades, extensive studies have been carried out on the elasticity of DHMS phases along the forst-erite–brucite join (Suzuki et al. 1983; Webb 1989; Zha et al. 1996, 1998; Fritzel and Bass 1997; Sinogeikin and Bass 1999; Sanchez-Valle et al. 2006; Jacobsen et al. 2008, 2009; Phan 2009; as well as this study). The density (ρ), elastic moduli (KS0 and G0, Voigt–Reuss–Hill averages), and acous-tic velocities (VP and VS) are summarized in Fig. 8a, b, and c, respectively, with linear regression fittings listed below:

The linear fits show that 1 wt% H2O reduces KS0 and G0 by 2.69(8) GPa (2 %) and 1.45(5) GPa (2 %), respec-tively, We also find that the isothermal bulk modulus (KT0) decreases by 2.71 GPa for every 1 wt% H2O added along the forsterite–brucite join (Ross and Crichton 2001; Holl et al. 2006). On the other hand, a linear relation for density versus water content is plotted in Fig. 8a. Addi-tion of 1 wt% H2O reduces ρ by 0.029(2) g/cm3, and

(12)KT =KS

1+ α · γ · T

(13)

ρ(g/cm3) = 3.27(2)− 0.029(2)× CH2O (R2 = 0.9767)

(14)

KS0(GPa) = 129.6(9)− 2.69(8)× CH2O (R2 = 0.9898)

(15)

G0(GPa) = 79.3(6)− 1.45(5)× CH2O (R2 = 0.9866)

(16)

VP(km/s) = 8.54(4)− 0.073(3)× CH2O (R2 = 0.9724)

(17)

VS(km/s) = 4.95(3)− 0.036(2)× CH2O (R2 = 0.9565)

linear regressions for moduli and velocities versus density (Birch’s law) could be fulfilled on the basis of Eq. (13).

In summary, we have presented new thermoelastic properties of OH-chondrodite, which was synthesized at 12 GPa and 1523 K, and coexists with OH-clinohumite and

Fig. 8 Elastic properties of DHMS along the forsterite–brucite join: a density as a function water content, b bulk modulus (K0) and shear modulus (G0) as a function of water content, c compressional (VP) and shear-wave (VS) velocities as a function of water content. Linear fits to the data (dashed lines) are given in the text (Eqs. 13–17). Ref-erences are given in Table 9

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hydrous olivine. The synthesis conditions correspond to the pressure–temperature condition around 360 km depth in a hydrous subducting slab. The chondrodite sample was characterized by multiple experimental methods: elec-tron microbe, Raman spectroscopy, single-crystal X-ray diffraction (153–787 K), and finally Brillouin spectros-copy. For DHMS phases along the forsterite–brucite joint, every 1 wt% increase in CH2O increases the mean thermal expansion coefficient (α0) by 3 %, decreases the mod-uli (KT0, KS0, and G0) by 2 %, and decreases the seismic velocities (VP and VS) by ~8 %. Hydration also enlarges the Grüneisen parameters for phase A, chondrodite, and clinohumite by 17–32 %, compared with anhydrous for-sterite. The elastic properties of OH-chondrodite found in this study are consistent with the previous isothermal compression studies of Fe,F-free samples. The thermal expansion coefficient α(T) at ambient pressure is con-strained by different models. The Fei equation yields: αV = 2.8(5) × 10−9 (K−2) × T + 40.9(7) × 10−6 (K−1) – 0.81(3) (K)/T2 (150–800 K); Suzuki and Kumar equations fit: V0 (0 K) = 365.8(3) Å3, Θ = 741 K, γMGD = 1.20(9)

[γ = 1.66(4) calculated from Eq. 11]. The thermoelastic properties reported in this study are important to anchoring more complete P–V–T equations of state for chondrodite and other DHMS phases in future studies at simultaneous high pressure and high temperature.

Acknowledgments This work was supported by US NSF grants EAR-1452344 (SDJ), EAR-1113369 (JRS), and EAR-114854 (TSD). SDJ also acknowledges support from the Carnegie/DOE Alliance Center (CDAC), and the David and Lucile Packard Foundation. Syn-thesis was performed at Bayerisches Geoinstitut, University of Bay-reuth, Germany, and supported in part by the Alexander von Hum-boldt Foundation.

Appendix 1

See Table 10.

Appendix 2

See Table 11.

Table 10 Anisotropic displacement parameters (Å2)

Atom U11 U22 U33 U23 U13 U12 Ueq

M1 0.0058(2) 0.0069(2) 0.0061(2) 0.0014(1) 0.0009(1) 0.0005(1) 0.0064(1)

M2 0.0085(2) 0.0065(2) 0.0071(2) 0.0029(1) −0.0004(1) 0.0004(1) 0.0072(1)

M3 0.0072(1) 0.0094(2) 0.0074(1) 0.0040(1) −0.0004(1) 0.0004(1) 0.0077(1)

Si 0.0042(1) 0.0056(1) 0.0054(1) 0.0022(1) 0.0000(0) 0.0000(0) 0.0050(1)

O1 0.0074(2) 0.0065(2) 0.0084(2) 0.0030(1) 0.0005(2) −0.0001(1) 0.0073(1)

O2 0.0068(2) 0.0074(2) 0.0070(2) 0.0015(2) 0.0002(1) −0.0001(1) 0.0073(1)

O3 0.0066(2) 0.0082(2) 0.0065(2) 0.0035(1) 0.0002(1) −0.0001(1) 0.0068(1)

O4 0.0043(2) 0.0084(2) 0.0075(2) 0.0027(2) 0.0001(1) 0.0000(1) 0.0067(1)

O5 0.0189(3) 0.0100(2) 0.0202(3) 0.0080(2) 0.0127(2) 0.0059(2) 0.0156(1)

H 0.054 (fixed)

Table 11 Unit-cell parameters of chondrodite as functions of temperature, as well as the principle expansion axes, which are normalized to the ones at 153 K

T (K) a (Å) b (Å) c (Å) α (°) V (Å3) X1/X10 X2/X20 X3/X30

153(2) 4.7402(3) 10.3341(8) 7.8899(8) 108.687(8) 366.12(4) 1 1 1

203(2) 4.7419(3) 10.3387(8) 7.8926(8) 108.693(8) 366.53(4) 1.00036 1.00030 1.00045

253(2) 4.7438(3) 10.3439(7) 7.8969(7) 108.698(7) 367.04(4) 1.00078 1.00076 1.00099

300(2) 4.7459(2) 10.3480(7) 7.9002(7) 108.702(7) 367.50(4) 1.00120 1.00114 1.00142

350(3) 4.7489(2) 10.3530(7) 7.9052(7) 108.706(8) 368.13(4) 1.00183 1.00164 1.00201

396(3) 4.7516(2) 10.3583(7) 7.9105(8) 108.708(9) 368.77(4) 1.00240 1.00217 1.00266

443(3) 4.7546(2) 10.3637(7) 7.9168(7) 108.715(9) 369.47(4) 1.00304 1.00267 1.00344

495(4) 4.7578(2) 10.3695(7) 7.9226(8) 108.722(9) 370.18(4) 1.00371 1.00318 1.00418

544(4) 4.7610(2) 10.3771(7) 7.9286(7) 108.735(9) 370.95(5) 1.00439 1.00380 1.00498

587(4) 4.7640(2) 10.3827(8) 7.9345(8) 108.737(8) 371.67(5) 1.00502 1.00435 1.00571

635(5) 4.7674(3) 10.3892(8) 7.9403(9) 108.750(9) 372.41(6) 1.00574 1.00486 1.00648

685(5) 4.7704(3) 10.3951(8) 7.9470(9) 108.764(9) 373.14(6) 1.00637 1.00534 1.00734

736(6) 4.7738(3) 10.4022(8) 7.9545(9) 108.771(9) 373.99(6) 1.00709 1.00599 1.00829

787(6) 4.7773(3) 10.4099(8) 7.9619(9) 108.795(9) 374.84(6) 1.00783 1.00653 1.00929

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